Town
Practice. Make use of the target dummies placed in your house and the town. Practice your new found arcana before you take them into the trials
Clothing. Your very first purchase should be a new robe. Your Damage and power in Wizard of legend comes from 3 sources. Arcana, relics and robe. You get the first 2 just by going through the museum. Spend your money on a new robe to get additional boost.
Invest. There are limits on relics and arcana you can take into the trials, those are lifted once inside. Having coins to spend on extra relics will help you a lot. So as soon as you can afford visit Jade, the lady that trades crystals for coins.

Dungeon
Purchases. You are limited to only 4 arcana when entering the trials however once inside you can have up to 6 active ones. Your first purchase should be at Virtuoso Iris, Arcana painter, to get one extra attack. No need to buy 2 since you should be getting one from the first boos soon enough. After that hit up Artisan Andres, the relic salesman for that extra boost to your abilities.
People you meet. Stay away from Nox the unfortunate giving away cursed relics. They often hinder you more than help. If you're running low on health and already purchased extra arcana you can trade them for health and a useful health related relic at Doctor Song. Nocturne the crdist is almost always a good choice. You will trade your regular arcana for an enhanced version of one.
Bosses. Bosses tend to attack in predictable patterns. Run around and avoid their attacks while they glow yellow. After a few stabs at you their glow will disappear and if hit they will be stunned letting you get in a few extra attacks.
Timing. You do not need to explore the whole map before summoning the boss. Sometimes it can be better to fight him early on when you still have full health. And after you win you can always go back and spend the cash he leaves behind.
Jumps. You can not fall into the pit just by running. Run all the way up to the edge and then jump to make safely across.